An oft-heard maxim among those scrambling for the line in the final days of this scandal–hit contest to host the 2018 and 2022 World Cups is that you can trust only those who say they will not vote for you. And given that almost all the voters, bar obvious exceptions, are giving little away, uncertainty and paranoia reign in the bars, hotel lobbies and meeting rooms of Zurich.

"With the exception of one or two people who have said they're committed whether out of affinity to a person or a country, we've not found people saying we cannot vote for you," the USA 2022 bid leader, Sunil Gulati, said. "If everyone had the number of votes they thought they had there would be far more than 22."

It is somehow emblematic of this opaque bidding race that only late last night came confirmation even on how many voters there will be, as Oceania accepted they could not replace the suspended Reynald Temarii with David Chung, leaving the decisions in the hands of 22 men.

It may seem unlikely, given the snow in both London and Zurich, but the priority for Prince William, David Beckham and David Cameron as they touched down in the Swiss city was fire-fighting.

The "three lions" – as they have been cringingly dubbed by the 2018 chief executive, Andy Anson – will have to walk a fine line in the coming days. If anything, they are more likely to resemble the three wise monkeys as they repeat England's key campaign messages while professing to see, hear and speak no evil.

In addition to the Concacaf president, Jack Warner, Cameron met the executive committee members Marios Lefkaritis and Mohamed bin Hammam, as well as taking part in rehearsals for Thursday's's presentation. He was planning a series of other meetings with voters tonight. As Cameron has already discovered, he will be asked incessantly over the coming days about the allegations against four of the 22 current executive committee members broadcast by Panorama.

But guided by a defiantly upbeat bid team, who believe they detect signs of a softening in support for the favourites Russia and harbour a belief that Spain/Portugal will struggle to extend their core vote beyond seven or eight they are fairly certain of, Cameron was instead accentuating the positive.

Cameron and Beckham spent two and a half hours at Fifa House outlining their message to Blatter and others on the day the Fifa president pulled down the shutters on any possibility of reopening an investigation into the damning Panorama allegations. Anson set the tone yesterday when he repeatedly referred to England as "underdogs" and sought to move on from the issues surrounding the British media in general and Panorama in particular. Beckham and Cameron added their weight to the message today.

The England bid team believe they have made headway in neutralising the impact of Panorama on their hoped for supporters – who include Jack Warner and Issa Hayatou. They now hope, perhaps optimistically, to focus on the positives of the England bid. They are using the boost given by Fifa's technical reports and a study commissioned from McKinsey on the economic value of each bid to frame the argument that theirs is objectively the best case. The idea that England will also provide the best spectacle and the best playing conditions for a Fifa that desperately needs a good World Cup on the pitch would also, on any objective analysis, play in their favour.

There is a clear choice between the jam today case of the English, the promise to open up new untapped markets of Russia and the politically astute alignment of the Spain/Portugal bid.

Russia's bid chairman and sports minister, Vitaly Mutko, today echoed the thoughts of the Iberian bid – who memorably declared "all the fish is sold" – when he said he believed all of his executive committee colleagues had made up their minds. "All I have is a whistle but after the match has begun I can do nothing. What we could do, we have done. There is not much we can do now."

Which begs the obvious question: can the "three lions" make any difference? The hope is not so much that they can sway the first preference votes of the voters but that they can consolidate existing support and influence the second and third preference votes that are almost certain to come into play. If a majority decision is not reached in the first round, the lowest-placed candidate drops out and the process continues until a winner is declared.

Anson said yesterday his team could see a "path to victory". That is believed to centre on ensuring Warner delivers his three votes to England and on securing the support of Turkey's Senes Erzik and the Japanese member Junji Ogura, in addition to that of England's Geoff Thompson.

There is less clarity around the intentions of Chung Mong-joon. It had also been unclear how Chung had planned to vote, although last night he accepted his ambition of being confirmed as the 23rd voter was over after Temarii said he would not resign from the executive committee.

England retain hopes that they too will add to their total if they can reach the second round. They then look to those supporting Holland/Belgium – thought to include Michel d'Hooge and Michel Platini – to switch to their cause and create a momentum that will see them through to a final showdown with Spain/Portugal. However, in an illustration of Gulati's point, Holland/Belgium today underlined their confidence that they could spring a surprise and knock out one of the favourites in the first round.

Blatter, expected to back Russia unless and until they are eliminated, is keen to avoid a Spain/Portugal and Qatar double for a range of reasons, not least because it would create even more suspicion around an already under-siege organisation, not to mention the damage to his re-election hopes next spring. So the hope is that he will swing behind England, along with some of the African voters, if it comes down to a face-off with Spain/Portugal.

The England team hope the swirl of speculation surrounding alleged collusion between the Spain/Portugal and Qatar bids will result in a marked reluctance among other bidders to join them.

As already well documented Warner, who today again attempted to shrug off the latest allegations against him, therefore remains vital to England's chances of progressing through the first round. If they can survive that, all bets are off. Which is why, for all the unease about whether England's prime minister and its future king should be kowtowing to Warner, Cameron's half-hour meeting could yet be key. It is also why all "three lions" will continue to dance on the head of a pin when faced with questions on Fifa corruption.